1. The Function $W(t) = 250 + 1200 E^{\frac{t}{2}}$ Models The Total Amount Of Waste Stored At A Landfill, Where $W(t$\] Is The Number Of Tons Of Waste, And $t$ Is The Number Of Years After 2012.a. How Many Tons Of Waste Are

1.1 Introduction

The function $W(t) = 250 + 1200 e^{\frac{t}{2}}$ is a mathematical model that represents the total amount of waste stored at a landfill. In this model, $W(t)$ is the number of tons of waste, and $t$ is the number of years after 2012. This function is an exponential function, which means that the rate of change of the waste is proportional to the current amount of waste. In this article, we will analyze the function and determine how many tons of waste are stored at the landfill at a given time.

1.2 Understanding the Function

The function $W(t) = 250 + 1200 e^{\frac{t}{2}}$ can be broken down into two parts: a constant term and an exponential term. The constant term, 250, represents the initial amount of waste stored at the landfill in 2012. The exponential term, $1200 e^{\frac{t}{2}}$, represents the amount of waste that is added to the landfill over time.

1.3 Analyzing the Exponential Term

The exponential term, $1200 e^{\frac{t}{2}}$, is the key to understanding how the waste at the landfill grows over time. The base of the exponential function, $e$, is a mathematical constant that is approximately equal to 2.718. The exponent, $\frac{t}{2}$, represents the number of years after 2012. As $t$ increases, the exponent also increases, causing the exponential term to grow rapidly.

1.4 Calculating the Amount of Waste

To calculate the amount of waste stored at the landfill at a given time, we can plug in the value of $t$ into the function $W(t) = 250 + 1200 e^{\frac{t}{2}}$. For example, if we want to know the amount of waste stored at the landfill in 2020, we can plug in $t = 8$ into the function.

1.5 Example Calculation

Let's calculate the amount of waste stored at the landfill in 2020.

$W(8) = 250 + 1200 e^{\frac{8}{2}}$ $W(8) = 250 + 1200 e^{4}$ $W(8) = 250 + 1200 \times 54.598$ $W(8) = 250 + 65512.8$ $W(8) = 65562.8$

Therefore, the amount of waste stored at the landfill in 2020 is approximately 65,562.8 tons.

1.6 Conclusion

In conclusion, the function $W(t) = 250 + 1200 e^{\frac{t}{2}}$ is a mathematical model that represents the total amount of waste stored at a landfill. By analyzing the function and plugging in the value of $t$, we can calculate the amount of waste stored at the landfill at a given time. In this article, we calculated the amount of waste stored at the landfill in 2020 and found that it is approximately 65,562.8 tons.

1.7 Future Work

Future work could involve analyzing the function further to determine the rate at which the waste is growing, and to identify any potential problems with the landfill. Additionally, the function could be modified to take into account other factors that may affect the amount of waste stored at the landfill, such as changes in population or waste management practices.

1.8 References

• [1] "Exponential Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/x2f-exponential-and-logarithmic-functions/x2f-exponential-functions/v/exponential-functions.

• [2] "Mathematical Modeling." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/science/mathematical-modeling.

• [3] "Waste Management." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/topic/waste-management.

1.9 Acknowledgments

The author would like to acknowledge the support of [insert name] in the preparation of this article.

2.1 Introduction

In the previous article, we analyzed the function $W(t) = 250 + 1200 e^{\frac{t}{2}}$ and determined how many tons of waste are stored at a landfill at a given time. In this article, we will answer some frequently asked questions about the function and provide additional information to help readers understand the topic.

2.2 Q&A

2.2.1 What is the initial amount of waste stored at the landfill?

The initial amount of waste stored at the landfill is represented by the constant term, 250, in the function $W(t) = 250 + 1200 e^{\frac{t}{2}}$. This means that in 2012, there were 250 tons of waste stored at the landfill.

2.2.2 How does the waste grow over time?

The waste grows over time due to the exponential term, $1200 e^{\frac{t}{2}}$, in the function $W(t) = 250 + 1200 e^{\frac{t}{2}}$. As $t$ increases, the exponent also increases, causing the exponential term to grow rapidly.

2.2.3 What is the rate of growth of the waste?

The rate of growth of the waste is proportional to the current amount of waste. This means that as the amount of waste increases, the rate of growth also increases.

2.2.4 How can we modify the function to take into account other factors that may affect the amount of waste stored at the landfill?

We can modify the function by adding or subtracting terms that represent the effects of other factors. For example, if we want to take into account changes in population, we can add a term that represents the effect of population growth on waste generation.

2.2.5 What are some potential problems with the landfill?

Some potential problems with the landfill include:

• Overcrowding: If the landfill becomes too full, it may not be able to accommodate additional waste.
• Environmental impact: The landfill may have a negative impact on the environment, such as pollution or habitat destruction.
• Health risks: The landfill may pose health risks to people living nearby, such as exposure to toxic chemicals.

2.2.6 How can we mitigate these problems?

We can mitigate these problems by:

• Implementing waste reduction and recycling programs to reduce the amount of waste generated.
• Improving waste management practices, such as using more efficient waste collection and disposal methods.
• Monitoring the landfill's environmental impact and taking steps to mitigate any negative effects.

2.3 Conclusion

In conclusion, the function $W(t) = 250 + 1200 e^{\frac{t}{2}}$ is a mathematical model that represents the total amount of waste stored at a landfill. By analyzing the function and answering frequently asked questions, we can gain a better understanding of the topic and identify potential problems with the landfill. We can also take steps to mitigate these problems and improve waste management practices.

2.4 Future Work

Future work could involve:

• Analyzing the function further to determine the rate at which the waste is growing.
• Identifying other factors that may affect the amount of waste stored at the landfill.
• Developing strategies to mitigate the problems associated with the landfill.

2.5 References

• [1] "Exponential Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/x2f-exponential-and-logarithmic-functions/x2f-exponential-functions/v/exponential-functions.

• [2] "Mathematical Modeling." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/science/mathematical-modeling.

• [3] "Waste Management." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/topic/waste-management.

2.6 Acknowledgments

The author would like to acknowledge the support of [insert name] in the preparation of this article.